A cylinder over a torus is by definition $S^1 \times S^1 \times I$ , here $I=[0,1]$.
One way to visualize it is to thick a torus in $\mathbb{R}^3$. ( $S^1 \times I$ is an annulus, and revolve it (the first $S^1$).) The other way is the compliment of a neighborhood of Hopf links in $S^3$. Suppose that the neiborhood lies in $\mathbb{R}^3 \subset \mathbb{R}^3 \cup {\infty}=S^3$. The second one is not actually a visualization since we (at least I) can't see the point at infinity.
We provide them with orientation induced by right hand orientation in $\mathbb{R}^3$ and $S^3$ respectively.
Question; So there is a orientation preserving homeomorphism between them but how does it look like? How can I visualize the transition from one to another? Especially I want to see how boundaries are transformed.